Sometime in the early Eighties, just off a flight at JFK airport, Arthur Miller went to ask for a limousine to take him home to Connecticut.

The boy at the counter was reading his play, The Crucible, and failed to order the car on time; when Miller gave his name to the desk again, a female attendant piped up: "Oh, that's the same name as the man who wrote that book." Miller replied, "Yeah, that's me," but the two of them now stared. "Oh no, that's impossible," she contradicted him. "He's been dead for years."

Arthur Miller lived to be almost 90 years old. His life was a mixture of incredible productivity and what looks like an enviable good time -- not only did he write Death of a Salesman, The Crucible, more than 20 other plays and numerous newspaper articles and short stories, he also married Marilyn Monroe, and after their separation (which was followed by her death) got married again, this time to a talented Austrian photographer called Inge Morath. He had several children (including a son, Daniel, who had Down syndrome). His daughter, Rebecca, herself a painter and film-maker, fell in love with Daniel Day-Lewis during the filming of The Crucible; they became engaged soon afterwards.

The very brilliance of Miller's life had downsides. In a way, he was haunted both by his past and his own longevity. For decades after their split, the press insisted on seeing him as the man who had slept with Marilyn and fulfilled that male fantasy. Nor was the airport taxi attendant the only woman he met who would assume he was already dead -- but more on that later.

Miller was an informed and passionate observer of the century that he lived through in all its momentousness and tragedy. In 1964, he travelled to Frankfurt to see the trial of administrators and guards from Auschwitz, expressing the blend of guilt and bewilderment that he felt in Incident at Vichy. He was elected president of the international writers' association Pen -- in part, it would later emerge, because the CIA wanted America to dominate western culture. In that role he campaigned on behalf of imprisoned writers across the world. He himself was a subject of interest to the US government, and the FBI kept a file on him for decades. He also became actively involved in US politics, and spoke out often against the Vietnam War.

After George W Bush became US president in 2000, taking Florida by 537 votes, Miller gave a speech in Prague. "We've just had an election ... And the man who won the election lost. And the man who lost the election is the winner. And we are asking people to be rational about life."

In March 2001, Miller delivered the Jefferson Lecture for the National Endowment for the Humanities, in which he criticised the US administration, even though he had received a phone call beforehand, implicitly requesting that he tone his message down; his reasoning, as he told his biographer, Christopher Bigsby, was that "I'm 85 and I don't care". Even in his latter years, he was political.

Miller believed his work would find an appreciative audience, but wrote that in America, "the favourite artist is the one who commits a masterpiece and then dies in the gutter, the sooner the better." He complied with the first part of this statement but not the second, and for much of his life he was treated as someone whose best days lay behind him. Critics trashed his later output, especially in his hometown of New York, where the New York Times -- so influential that a bad review could close a production down -- slammed play after play. His autobiography, Timebends, received the worst response of all. When it came out in 1987, David Denby wrote in the New Republic that it had "an amorphous and sluggish feel to it". In case that wasn't enough, he summed up Miller's portrayal of Monroe as "mating with the golden-haired dream, quite a treat for a Jewish boy".

Often it is women who are defined by their relationships with famous men, but in Miller's case, he remained linked to Monroe in the public imagination, so much so that his publishers wanted to put a picture of the blonde star, rather than Miller himself, on his memoir's front cover. At a celebration of his 80th birthday, a reporter asked him if he still dreamt of Monroe; his response was to swing a punch at the man.

This is the second instalment in Bigsby's biography of Miller. Running to 520 pages (excluding notes and index) it takes up just after his split with Monroe, as he begins his much more stable and lasting relationship with Morath. Bigsby's research is thorough and he has crafted the book with care, but much of it lacks momentum. Perhaps that's because Miller's political commitments, while impressive, don't have the must-read lure of personal intrigue. Bigsby reveals little about the affairs that Miller did have. He mentions one that lasted for a year and hints at others. It would have been enlightening to have more details, and might have more shed light on the playwright's complex nature. For "the theme of betrayal runs throughout Miller's work", Bigsby says; and "guilt was in some way a motor force".

Surprisingly, the story gains a racy element at the very end of Miller's life. After his second wife's death he met a young woman called Agnes Barley at a dinner party. She didn't know much about theatre, and later she admitted that she had thought he was dead. But they hit it off and quickly became intimate, despite a 55-year age gap (Barley was 32). He and Barley moved in together, and he left her his New York apartment in his will. When a New York Times reporter enquired about their relationship Miller would only say: "I like the company of women ... Life is very boring without them."

As for Barley, she truly loved him, it seems -- at least, five years on, the answering machine still has his voice.